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I want to know the meaning of ≡ this symbol. I read about it on Wikipedia but didn't actually understand what that means.

I saw this symbol in coordinate geometry where it was given that $A \equiv (1,2)$. But what's wrong with $A = (1, 2)$? Is it wrong? Sometimes, my teacher also represents coordinates as $A(1, 2)$.

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    It can mean "is defined to be equal to" (for which another common symbol is $:=$). Another meaning is "is identically equal to" (for example we write $\sin^2x+\cos^2x\equiv 1$ to emphasize that this equation holds for all $x\in\Bbb{R}$ (actually for all $x\in\Bbb{C}$ as well)). It can also mean congurence as in modular arithmetic. The intended meaning depends on context, but as for your concern, there's nothing wrong with saying $A=(1,2)$ with some remark like "We define $A$ to be $A=(1,2)$" or "Consider the point $A=(1,2)$" preceding it. – peek-a-boo Feb 12 '22 at 12:40
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    What is "the" article you read on Wikipedia? – Blue Feb 12 '22 at 12:53
  • in logic $A\equiv B$ usually means that the string of symbols $A$ and $B$ represent the same thing, from there it comes the use in mathematics to say that $A$ and $B$ represents the same thing, what can be also understood as $A$ is defined to represent $B$, when $B$ have an already known meaning – Masacroso Jun 08 '22 at 08:46

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As far as my understanding can be trusted, $\equiv$ is often used to draw the reader's attention on the nature of the objects being considered. There is always a notion of equality somewhere behind it, but that is more subtle than a simple equality and highly dependent on the context.

In the case of plane geometry, I interpret $A \equiv (1,2)$ as "$A$ is the point of the plane given by the coordinates $(1,2)$ relatively to the considered coordinate system". There is a bijection between the set of points on the plane and the set of couples of real numbers for a given coordinate system, but a point is not a couple of real numbers.

In arithmetics, we say that $a \equiv b \mod c$ when $a$ and $b$ are congruent. Going further in that direction, it means that $a$ and $b$ represent the same element in the group $\mathbb{Z}/c\mathbb{Z}$, but as element of $\mathbb{Z}$, they might be different.

A third common example is when we write $f \equiv 0$ to say that $f$ is identically $0$. $f$ is not the constant $0$, but rather the constant $0$ function.

nicomezi
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